Thermal excitation of flexoelectricity in silicon

Flexoelectricity, an electromechanical coupling between strain gradient and polarization, offers a promising dimension to enrich silicon-based devices. Although the flexoelectricity of silicon is known, some fundamental aspects remain ambiguous, such as the discrepancy between experimental results and theoretical predictions, the influence of doping concentration, and the role of the band gap. Here, we measured the flexoelectricity of intrinsic and heavily doped Si over the temperature range of $\ensuremath{-}50--200{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$. The flexoelectric coefficient is of \ensuremath{\sim}2.6 \textmu{}C/m and barely varies with temperature in doped silicon, while in intrinsic silicon, it varies by nearly two orders of magnitude from \ensuremath{\sim}15.2 nC/m to 1.8 \textmu{}C/m as temperature increases. We show that their different temperature dependencies correspond to the temperature-insensitive donor ionization in doped silicon and the temperature-sensitive intrinsic excitation in intrinsic silicon, with the latter captured by a quantitative relationship between flexoelectricity, temperature, and band gap. Furthermore, similar experimental results on germanium (Ge) suggest the universality of this relationship in first-generation semiconductors. These findings would offer valuable reference for developing Si-based electromechanical devices as well as understanding the strain-gradient effects on semiconductor band structures (flexoelectronics).